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Metric Tensor, Connection, Curvature and Geodesic In this chapter we give a brief introduction to the elements of differential geometry as used in general relativity. The main motivation that led Einstein to develop general relativity was the equivalence principle. In all forces except gravity the acceleration of a particle depends on its (inertial) mass, the lighter the particle is the more it will accelerate in the same field. For example a particle in an electric field has acceleration a = mq E. Here q is the strength with which the particle couples to the electric field also called the electric charge but m is the inertia of the particle. Thus lighter particles will accelerate more. But for gravity the relation is a = qgrav g, here g is the strength m of the gravitational field (acceleration due to gravity) and qgrav is the strength with which the particle couples to the gravitational field. According to the equivalence principle, this is proportional to the inertial mass m. This means qgrav /m is a universal constant usually taken to be unity. Thus all particles accelerate with the same acceleration g. According to legend, this was demonstrated by Galileo in his famous leaning tower of Pisa experiments. This led Einstein to suspect that gravity was very different from all the other forces. Since all particle behave the same in gravity, Einstein said that gravity simply changes the geometry of space and time and all particle then move freely in this curved space time. It is as if the stage is modified and all actors have to act on the same stage. With electromagnetic force the path travelled by a particle depends on its mass. Whereas for gravity it does not. So Einstein said that if gravity is the only force present then all particles will follow the same straight path (the shortest route) but in a curved space time. Thus we have to understand how matter curves space and time. For this we have to understand how to describe geometry of four-dimensional space times. I. METRIC TENSOR We know that in special relativity in cartesian coordinates, the distance that is invariant under both ordinary rotations in space and Lorentz transformations (rotation that mixes space and time variables) is ds2 = c2 dt2 − dx2 − dy 2 − dz 2 = ηµν dxµ dxν This simple form is only valid in Cartesian coordinates. In other orthogonal coordinates systems it will look like, ds2 = c2 dt2 − h1 dr2 − h2 dθ2 − h3 dφ2 = gµν dxµ dxν where h1 = 1, h2 = r2 and h3 = r2 sin2 θ and if x = (ct, r, θ, φ) then g00 = 1, g11 = h1 , g22 = h2 , g33 = h3 . One can also work in non-orthogonal coordinate systems. For example, we could choose two linearly independent vectors ê1 and ê2 that are not mutually orthogonal to describe the position of a particle in two space dimensions. ~r = x1 ê1 + x2 ê2 . Now if we calculate the square of the distance between two neighboring points ds2 = d~r · d~r, we get d~r = dx1 ê1 + dx2 ê2 , hence ds2 = d~r · d~r = (dx1 )2 + (dx2 )2 + 2(ê1 · ê2 ) dx1 dx2 . If the two unit vectors were orthogonal then there would be no cross term like dx1 dx2 in the distance expression. For a general non-orthogonal coordinate system then all kinds of cross terms would also be present. Thus the most general case is that of non-orthogonal and curvilinear coordinates which means the metric (expression for distance squared) has the general form, ds2 = gµν (x)dxµ dxν (1) 2 Now we can verify many properties of the metric. First we have to define the covariant coordinate xµ . In special relativity it is dxµ = ηµν dxν , where ηµν = diag(1, −1, −1, −1), since this is sufficient to ensure that dxµ dxµ = c2 dt2 − dx2 − dy 2 − dz 2 is a Lorentz invariant. In general relativity, we look for four-distance such as ds2 that are invariant under general coordinate transformations including Lorentz transformation. Thus we should write, ds2 = gµν (x)dxµ dxν = dxµ dxµ This means dxµ = gµν (x)dxν . Any object V µ (x) that transforms like dxµ under coordinate transformations is a contra-variant vector. Any object Vµ (x) that transforms like dxµ under coordinate transformations is a co-variant vector. Homework : (a) Find ds2 on a torus of small radius a and big radius b. (b) Find ds2 on an 2 2 ellipsoid of revolution (the surface obtained by rotating the ellipse xa2 + yb2 = 1 about the x-axis). A. Equation of a Geodesic from the Metric In an earlier chapter we showed that the lagrangian of a free particle in relativistic physics is, L = −mc v u u 2t 2 ~r˙ 1− 2 c we can calculate the action as, S= Now we write cdt = √ Z tf ti v u Z tf u t 2 L dt = −mc ti c2 dt2 and take it inside the square root. S = −mc But 2 ~r˙ 1 − 2 dt c Z tf q ti c2 dt2 − (d~r)2 q c2 dt2 − (d~r)2 = ds. Thus, S = −mc Z f i ds Thus the action of a free particle is proportional to the four dimensional distance between the initial and final points (events). The principle of least action is same as finding the extremum of the distance between two points in four dimensions (one must note that c2 dt2 − (d~r)2 may be positive, negative or zero, hence ds, the distance between events is not always real). Hence the shortest possible route between two points in space-time is also the path that minimizes the action and hence is the path of the particle. Now I want to repeat this exercise in curved space-time where the distance between points is given by Eq.(1). The idea now is that I choose a clever way of parametrizing the path. I choose s itself as the parameter. This is similar to describing the circle x2 + y 2 = a2 using x = a cos(s/a) and y = a sin(s/a) where s is the distance along the circle s = aθ. Hence I will describe the path as xµ (s). Thus, S = −mc Z f i ds = −mc Z f ds2 i ds = −mc Z f gµν (x)dxµ dxν i ds = −mc Z f i gµν (x) dxµ dxν ds ds ds 3 We call dxν ds = ẋν (s). Then, S = −mc Z f i µ ν gµν (x)ẋ (s)ẋ (s) ds = Z f i L(x, ẋ) ds where the lagrangian is given by L(x, ẋ) = −mc gµν (x)ẋµ (s)ẋν (s). Thus the path that minimizes the action is that special path xµ (s) = cµ (s), that obeys the lagrange equation. d ∂L(x, ẋ) ∂L(x, ẋ) = ρ ds ∂ ẋ ∂xρ (2) After calculating the required derivatives of the lagrangian we get, −2mc d ∂gµν (c(s)) µ (gρν (c(s))ċν (s)) = −mc ċ (s)ċν (s) ds ∂cρ (3) But, dgρν (x) = ẋµ (s)gρν,µ (x) ds Here the notation means f,ν (x) ≡ ∂f (x) . ∂xν Or, 2 ċν (s)ċσ (s)gρν,σ (c(s)) + 2 gρν (c(s))c̈ν (s) = gσν,ρ (c(s))ċσ (s)ċν (s) (4) Now we show that, g µρ gρν = δνµ . To see this we first note that for any vector V µ and Vµ , V µ = g µν Vν . Choose Vν = gνρ aρ and V µ = g µρ aρ hence g µρ aρ = g µν gνρ aρ . This means g µρ = g µν gνρ . It is easy to see that g µρ = δ µρ since for any vector V µ = g µρ V ρ = δ µρ V ρ . Hence δ µν = g µρ gρν . We multiply both sides of Eq.(4) by g µρ and divide by two to get, c̈µ (s) + Γµνσ (c(s))ċν (s)ċσ (s) = 0 (5) 1 1 Γµνσ (x) = g µρ (x)gρν,σ (x) − g µρ (x)gσν,ρ (x) = g µρ (x)(gρν,σ (x) + gνρ,σ (x) − gσν,ρ (x)) 2 2 (6) where, Homework : Find the objects Γ for the metric of points on a sphere (a is the radius of the sphere). ds2 = a2 dθ2 + a2 sin2 (θ)dφ2 From this find the equation of the geodesic on the surface of the sphere. II. CONNECTION Independent of the metric concept, one can introduce a concept called connection. Suppose I have a vector field : V µ (x), that is a function at each point in space-time whose value is a vector (µ-th 0 component is chosen for illustration). Then one can ask if the two vector V (x) and V (x ) are parallel 0 at different points x and x ? In general the answer is no as the vector field can twist and turn as it moves from point to point. But we can ask ourselves if there is some way we can move a vector 0 0 from x to x such that the vector Ṽ (x ) is now parallel to V (x). To do this we need a concept known as connection. Connection is a quantity that tells us how to find a vector at some other point that is parallel to a given vector at the starting point. To do this consider two neighboring points x and x + dx (remember that x means (ct, x, y, z) or (ct, r, θ, φ) or anything with four entries). We want 4 Ṽ µ (x + dx) to be parallel to V µ (x). Let us now focus on the difference Ṽ µ (x + dx) − V µ (x), as dx → 0, Ṽ → V , this is because as the points come closer and closer, the two vectors should coincide : Ṽ → V . Thus for closely separated points the difference between these two vectors should be proportional to the separation dx. Ṽ µ (x + dx) − V µ (x) = dxν Hνµ (V (x)) (7) where Hνµ (V (x)) is some coefficient. Now consider two such vectors V and W . Then we have, W̃ µ (x + dx) − W µ (x) = dxν Hνµ (W (x)) (8) Since V, W are vectors, they have to respect the principle of superposition. This means the same relation should also hold for Z = V + W . Thus, Z̃ µ (x + dx) − Z µ (x) = dxν Hνµ (Z(x)) (9) Adding Eq.(7) and Eq.(8) and subtracting from Eq.(9) we get, Hνµ (U (x) + V (x)) = Hνµ (U (x)) + Hνµ (V (x)) (10) Hνµ (V (x)) = Γµνρ (x)V ρ (x) (11) This is possible only if, Here Γµνρ (x) are known as the coefficients of a connection. They are more commonly known as Christoffel symbols. If someone tells me what these are for some space-time only then I know how to transport a vector parallel to itself from point to point. Hence, Ṽ µ (x + dx) − V µ (x) = dxν Γµνρ (x)V ρ (x) A. (12) Equation of the geodesic from the Connection In three dimensional space a curve is described by ~r(t). The vector tangent to this curve is the velocity vector ~r˙ (t). Similarly, in four dimensions if the curve is cµ (s), the vector tangent to this d µ c (s). Now if I transport a vector that is tangent to a curve along the curve curve is ċµ (s) = ds parallel to itself to another point on the curve, it will remain tangent to the curve provided the curve is the straightest possible curve namely the geodesic. For this I choose the vector field, V µ (x) = Z f i 0 0 0 ds ċµ (s ) δ 4 (x − c(s )) (13) where δ 4 (x − c(s)) = δ(x0 − c0 (s))δ(x1 − c1 (s))δ(x2 − c2 (s))δ(x3 − c3 (s)). This vector field has the property that it represents the tangent to the geodesic c when x lies on the curve and is zero otherwise. The vector Ṽ µ (x + dx) represents the vector that is parallel to V µ (x) but lives at point x + dx. But we have argued that this vector is also tangent to the curve since V µ was a tangent vector to begin with. Hence, Ṽ µ (x + dx) = V µ (x + dx). Substituting this into Eq.(12) we get, V µ (x + dx) − V µ (x) = dxν ∂V µ (x) = dxν Γµνρ (x)V ρ (x) ∂xν (14) We know from the definition that the vector V (x) is zero unless x lies on the geodesic c. For the same reason x + dx also lies on the geodesic. Hence there exists a parameter s such that x = c(s). This means s = c−1 (x). As x changes from one value to the next along the geodesic c, this parameter s 5 also changes. Now we divide both sides of Eq.(14) with ds. Then we substitute Eq.(13) into Eq.(14) to get, Z f i Z f dxν ∂ 4 dxν µ 0 0 0 0 ds ċ (s ) δ (x − c(s )) = ds ċρ (s ) δ 4 (x − c(s )) Γνρ (x) ν ds ∂x ds i 0 0 µ ν (15) ν d ∂ ∂ d Now consider the identity ds f (x(s)) = dx f (x), or, dx = ds . Also the delta function forces ds ∂xν ds ∂xν 0 0 −1 x = c(s ) but we have argued that x = c(s) for a value of s = c (x). This means s = s . Hence ν d = dsd 0 . On the right hand side of Eq.(15) we may substitute dx = ċν (s) and x = c(s). Since x lies ds ds on the geodesic c. After substituting all this into Eq.(15) we get, Z f i Z f d 4 0 0 0 0 ν µ ds ċ (s ) 0 δ (x − c(s )) = ċ (s) Γνρ (c(s)) ds ċρ (s ) δ 4 (x − c(s )) ds i 0 0 µ (16) Now we may integrate the left hand side by parts, to get, Z f i 0 0 ds ċµ (s ) Z f h is=sf d 4 d 0 0 0 0 0 µ 0 4 δ (x − c(s )) = ċ (s ) δ (x − c(s )) − ds δ 4 (x − c(s )) 0 ċµ (s ) 0 s=si ds ds i (17) Here c(sf ) = xf is the end point of the curve (geodesic) and c(si )xi is the starting point. We assume that x is on this curve but somewhere in between xi and xf so that x 6= c(si ), c(sf ). Thus the boundary term drops out and we get, Z f i 2 µ Z f d 4 d2 cµ (s ) 0 0 0 4 δ (x − c(s )) = − ds δ (x − c(s )) ds0 ds0 2 i 0 0 0 ds ċµ (s ) 0 (18) c (s ) . Substituting Eq.(18) into Eq.(16) we obtain keeping in mind that s = s since dsd 0 ċµ (s ) = d ds 02 and equating the coefficients of the delta functions on both sides we get, 0 − d2 cµ (s) = Γµνρ (c(s)) ċν (s)ċρ (s) ds2 0 (19) This equation Eq.(19) is the same as Eq.(5). Thus the Γ found in Eq.(5) written in terms of the metric gµν is nothing but the geometric concept of a connection used to define parallel transport of vectors.